A game in normal form consists of: Set of players = {1, , } - PowerPoint PPT Presentation

T RUTH J USTICE A LGOS Game Theory I: Basic Concepts Teachers: Ariel Procaccia (this time) and Alex Psomas

NORMAL-FORM GAME • A game in normal form consists of: ◦ Set of players 𝑂 = {1, … , 𝑜} ◦ Strategy set 𝑇 ◦ For each 𝑗 ∈ 𝑂 , utility function 𝑣 𝑗 : 𝑇 𝑜 → ℝ : if each j ∈ 𝑂 plays the strategy 𝑡 𝑘 ∈ 𝑇 , the utility of player 𝑗 is 𝑣 𝑗 (𝑡 1 , … , 𝑡 𝑜 )

THE PRISONER’S DILEMMA • Two men are charged with a crime • They are told that: ◦ If one rats out and the other does not, the rat will be freed, other jailed for nine years ◦ If both rat out, both will be jailed for six years • They also know that if neither rats out, both will be jailed for one year

THE PRISONER’S DILEMMA Cooperate Defect -1,-1 -9,0 Cooperate Defect 0,-9 -6,-6 What would you do?

ON TV http://youtu.be/S0qjK3TWZE8

THE PROFESSOR’S DILEMMA Class Listen Sleep 10 6 ,10 6 -10,0 Professor Make effort Slack off 0,-10 0,0 Dominant strategies?

NASH EQUILIBRIUM • In a Nash equilibrium, no player wants to unilaterally deviate • E ach player’s strategy is a best response to strategies of others • Formally, a Nash equilibrium is a vector of strategies 𝒕 = 𝑡 1 … , 𝑡 𝑜 ∈ 𝑇 𝑜 such ′ ∈ 𝑇, that for all 𝑗 ∈ 𝑂, 𝑡 𝑗 ′ , 𝑡 𝑗+1 , … , 𝑡 𝑜 ) 𝑣 𝑗 𝒕 ≥ 𝑣 𝑗 (𝑡 1 , … , 𝑡 𝑗−1 , 𝑡 𝑗

THE PROFESSOR’S DILEMMA Class Listen Sleep 10 6 ,10 6 -10,0 Professor Make effort Slack off 0,-10 0,0 Nash equilibria?

ROCK-PAPER-SCISSORS R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0 Nash equilibria?

MIXED STRATEGIES • A mixed strategy is a probability distribution over (pure) strategies • The mixed strategy of player 𝑗 ∈ 𝑂 is 𝑦 𝑗 , where 𝑦 𝑗 (𝑡 𝑗 ) = Pr[𝑗 plays 𝑡 𝑗 ] • The utility of player 𝑗 ∈ 𝑂 is 𝑜 𝑣 𝑗 𝑦 1 , … , 𝑦 𝑜 = ෍ 𝑣 𝑗 𝑡 1 , … , 𝑡 𝑜 ⋅ ෑ 𝑦 𝑘 (𝑡 𝑘 ) (𝑡 1 ,…,𝑡 𝑜 )∈𝑇 𝑜 𝑘=1

EXERCISE: MIXED NE 1 1 • Exercise: player 1 plays 2 , 2 , 0 , player 2 1 1 plays 0, 2 , 2 . What is 𝑣 1 ? 1 1 1 • Exercise: Both players play 3 , 3 , 3 . What is 𝑣 1 ? R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0

EXERCISE: MIXED NE Poll 1 ? Which is a NE? 1 1 1 1 1 1 1 1 1 1 1. 2 , 2 , 0 , 2 , 2 , 0 3. 3 , 3 , 3 , 3 , 3 , 3 1 1 1 1 1 2 2 1 2. 2 , 2 , 0 , 2 , 0, 4. 3 , 3 , 0 , 3 , 0, 2 3 R P S R 0,0 -1,1 1,-1 P 1,-1 0,0 -1,1 S -1,1 1,-1 0,0

NASH’S THEOREM • Theorem [Nash, 1950]: In any (finite) game there exists at least one (possibly mixed) Nash equilibrium • What about computing a Nash equilibrium? Stay tuned…

DOES NE MAKE SENSE? • Two players, strategies are {2, … , 100} • If both choose the same number, that is what they get • If one chooses 𝑡 , the other 𝑢 , and 𝑡 < 𝑢 , the former player gets 𝑡 + 2 , and the latter gets 𝑡 − 2 • Poll 2: What would you choose? 95 96 97 98 99 100

CORRELATED EQUILIBRIUM • Let 𝑂 = {1,2} for simplicity • A mediator chooses a pair of strategies (𝑡 1 , 𝑡 2 ) according to a distribution 𝑞 over 𝑇 2 • Reveals 𝑡 1 to player 1 and 𝑡 2 to player 2 • When player 1 gets 𝑡 1 ∈ 𝑇 , he knows the distribution over strategies of 2 is Pr 𝑡 2 𝑡 1 = Pr 𝑡 1 ∧ 𝑡 2 = 𝑞 𝑡 1 , 𝑡 2 Pr 𝑡 1 Pr[𝑡 1 ]

CORRELATED EQUILIBRIUM ′ ∈ 𝑇 • Player 1 is best responding if for all 𝑡 1 ′ , 𝑡 2 ) ෍ Pr 𝑡 2 𝑡 1 𝑣 1 𝑡 1 , 𝑡 2 ≥ ෍ Pr 𝑡 2 𝑡 1 𝑣 1 (𝑡 1 𝑡 2 ∈𝑇 𝑡 2 ∈𝑇 • Equivalently, ′ , 𝑡 2 ) ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 1 𝑡 1 , 𝑡 2 ≥ ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 1 (𝑡 1 𝑡 2 ∈𝑇 𝑡 2 ∈𝑇 • 𝑞 is a correlated equilibrium (CE) if both players are best responding • Every Nash equilibrium is a correlated equilibrium, but not vice versa

GAME OF CHICKEN http://youtu.be/u7hZ9jKrwvo

GAME OF CHICKEN • Social welfare is the sum of utilities Dare Chicken • Pure NE: (C,D) and 0,0 4,1 Dare (D,C), social welfare = 5 • Mixed NE: both 1,4 3,3 (1/2,1/2), social Chicken welfare = 4 • Optimal social welfare = 6

GAME OF CHICKEN • Correlated equilibrium: ◦ (D,D): 0 Dare Chicken 1 ◦ (D,C): 0,0 4,1 Dare 3 1 ◦ (C,D): 3 1,4 3,3 Chicken 1 ◦ (C,C): 3 16 • Social welfare of CE = 3

IMPLEMENTATION OF CE • Instead of a mediator, use a hat! • Balls in hat are labeled with “chicken” or “dare”, each blindfolded player takes a ball Poll 3 ? Which balls implement the distribution of the previous slide? 1. 1 chicken, 1 dare 3. 2 chicken, 1 dare 2. 1 chicken, 2 dare 4. 2 chicken, 2 dare

CE AS LP • Can compute CE via linear programming in polynomial time! find ∀𝑡 1 , 𝑡 2 ∈ 𝑇, 𝑞 𝑡 1 , 𝑡 2 ′ ∈ 𝑇, s.t .t. ∀𝑡 1 , 𝑡 1 ′ , 𝑡 2 ) ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 1 𝑡 1 , 𝑡 2 ≥ ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 1 (𝑡 1 𝑡 2 ∈𝑇 𝑡 2 ∈𝑇 ′ ∈ 𝑇, ∀𝑡 2 , 𝑡 2 ′ ) ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 2 𝑡 1 , 𝑡 2 ≥ ෍ 𝑞 𝑡 1 , 𝑡 2 𝑣 2 (𝑡 1 , 𝑡 2 𝑡 1 ∈𝑇 𝑡 1 ∈𝑇 ෍ 𝑞 𝑡 1 , 𝑡 2 = 1 𝑡 1 ,𝑡 2 ∈𝑇 ∀𝑡 1 , 𝑡 2 ∈ 𝑇, 𝑞 𝑡 1 , 𝑡 2 ∈ [0,1]